Validity of $\exists x \forall y: x \le y$ in set of integers and set of natural numbers

Question

If I say $\exists x\in \mathbb{N} $ such that $\forall y \in \mathbb{N},$ $x \leq y$, this is a true statement. I am assuming that the set of natural numbers is defined for $0\leq x < \infty $.

But if I say the same thing for the set of all integers: $\exists x\in \mathbb{Z} $ such that $\forall y \in \mathbb{Z},$ $x \leq y$ is a false statement, and I need help with my wording here: It is false because there does not exist an integer $x$ such that there are no integers less than $x$.

Am I correct in my logic here?

Answer 1

The statement is saying that there is a smallest element. This is true for natural numbers, false for integers. [...] [Your] claims are correct.

-- Ted (in a comment).

Answer 2

Well it seems that your logic is false but you have to say why do you think there is no such number as x. Cause it exists .